Many applications of signal processing require the sampling of a non-bandwidth limited signal. As is well known in the art, an aliasing error will be introduced if the Nyquist criterion, i.e., the relationship EQU .pi./T.ltoreq..omega..sub.max
is not satisfied, where T is the sampling interval and .omega..sub.max is the upper frequency limit of a sampled signal. Thus, any accurate reconstruction of the sampled signal utilizing the erroneous sampled values would be impossible.
A system for sampling a steady-state, non-bandwidth limited signal without aliasing is described in a co-pending, commonly assigned patent application to the same inventor entitled "A System For Converting Analog Signals To A Discrete Representation Without Aliasing" Ser. No. 036,763 Apr. 9, 1987. This application is hereby incorporated by reference.
However, many signals of interest include transient components, e.g., decaying exponentials, and are not suitable for analysis by the above-described system.
One example of such a transient signal is the impulse response of a linear system. A linear system is characterized by its natural resonances that are of the form ##EQU2## where the quantities s.sub.i are the natural frequencies of the system and the quantities r.sub.n,i are the resonant amplitudes of the frequency components. Additionally, the quantities s.sub.i and r.sub.n,i are the poles and residues, respectively, of the frequency transform function, H(s), of the linear system. Succeeding terms in the above expression represent higher order resonances at the same natural frequency s.sub.i.
Thus, the quantities s.sub.i and r.sub.n,i are s-plane parameters describing the continuous analog signal.
A system containing single order poles only, has an impulse response EQU h.sub.i (t)=.SIGMA.r.sub.i e.sup.s.sbsp.i.sup.t
which is characteristic of the system and contains the necessary information for describing the poles and residues of the network. For the sake of simplicity in presentation and without any real loss in generality, systems with only single order poles are considered.
Methods have been developed for estimating the parameters s.sub.i and r.sub.i from sampled values of h(t) taken .tau. seconds apart. According to the Nyquist theorem, these sampled values closely approximate h(t) if the spectral energy of h(t) is essentially limited to a frequency less than f.sub.m =1/2.tau..
Any physical system has an infinite bandwidth because each resonance (pole) has a frequency response that drops off at a finite rate. Therefore, any finite sampling rate will violate the Nyquist criterion thereby causing aliasing errors to prevent an accurate determination of s.sub.i and r.sub.i.
An accurate determination of the s-plane parameters is important for a number of reasons. These parameters define the impulse response of the system. The impulse response determines the response of the system to transient test stimuli, such as step functions, and thus its characterization is critical in high speed testing environments. Secondly, the s-plane parameters define the transfer function of the system. Accurate measurement of the s-plane parameters would facilitate a compensation system for a less than ideal transfer characteristic of a physical system.
Accordingly, a system for sampling an analog signal, including a transient component, without aliasing is greatly needed in the signal processing art. The resulting samples would be of great utility in reconstructing the analog signal.